Cantor distribution

Short description: Probability distribution

Cantor

Cumulative distribution function
Parameters	none
Support	Cantor set, a subset of [0,1]
pmf	none
CDF	Cantor function
Mean	1/2
Median	anywhere in [1/3, 2/3]
Mode	n/a
Variance	1/8
Skewness	0
Kurtosis	−8/5
MGF	$e^{t/2}\prod _{k=1}^{\mathrm{\infty }}\cosh \left(\frac{t}{3^k}\right)$
CF	$e^{it/2}\prod _{k=1}^{\mathrm{\infty }}\cos \left(\frac{t}{3^k}\right)$

The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.

This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution.

Its cumulative distribution function is continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the Devil's staircase, although that term has a more general meaning.

Characterization

The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets:

$\begin{array}{rl}{C}_0=& [0,1]\\ C_1=& [0,1/3]\cup [2/3,1]\\ C_2=& [0,1/9]\cup [2/9,1/3]\cup [2/3,7/9]\cup [8/9,1]\\ C_3=& [0,1/27]\cup [2/27,1/9]\cup [2/9,7/27]\cup [8/27,1/3]\cup \\ & [2/3,19/27]\cup [20/27,7/9]\cup [8/9,25/27]\cup [26/27,1]\\ C_4=& [0,1/81]\cup [2/81,1/27]\cup [2/27,7/81]\cup [8/81,1/9]\cup [2/9,19/81]\cup [20/81,7/27]\cup \\ & [8/27,25/81]\cup [26/81,1/3]\cup [2/3,55/81]\cup [56/81,19/27]\cup [20/27,61/81]\cup \\ & [62/81,21/27]\cup [8/9,73/81]\cup [74/81,25/27]\cup [26/27,79/81]\cup [80/81,1]\\ C_5=& \cdots \end{array}$

The Cantor distribution is the unique probability distribution for which for any C_t (t ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in C_t containing the Cantor-distributed random variable is identically 2^−t on each one of the 2^t intervals.

Moments

It is easy to see by symmetry and being bounded that for a random variable X having this distribution, its expected value E(X) = 1/2, and that all odd central moments of X are 0.

The law of total variance can be used to find the variance var(X), as follows. For the above set C₁, let Y = 0 if X ∈ [0,1/3], and 1 if X ∈ [2/3,1]. Then:

$\begin{array}{rl}\mathrm{var}(X)& =\mathrm{E}(\mathrm{var}(X\mid Y))+\mathrm{var}(\mathrm{E}(X\mid Y))\\ & =\frac{1}{9}\mathrm{var}(X)+\mathrm{var}\left\{\begin{array}{cc}1/6& \text{with probability}1/2\\ 5/6& \text{with probability}1/2\end{array}\right\}\\ & =\frac{1}{9}\mathrm{var}(X)+\frac{1}{9}\end{array}$

From this we get:

$\mathrm{var}(X)=\frac{1}{8}.$

A closed-form expression for any even central moment can be found by first obtaining the even cumulants

${\kappa }_{2n}=\frac{2^{2n-1}(2^{2n}-1)B_{2n}}{n(3^{2n}-1)},$

where B_2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.

Further reading

• Hewitt, E.; Stromberg, K. (1965). Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer-Verlag. https://archive.org/details/realabstractanal00hewi_0.  This, as with other standard texts, has the Cantor function and its one sided derivates.
• Hu, Tian-You; Lau, Ka Sing (2002). "Fourier Asymptotics of Cantor Type Measures at Infinity". Proc. AMS 130 (9): pp. 2711–2717.  This is more modern than the other texts in this reference list.
• Knill, O. (2006). Probability Theory & Stochastic Processes. India: Overseas Press. 
• Mattilla, P. (1995). Geometry of Sets in Euclidean Spaces. San Francisco: Cambridge University Press.  This has more advanced material on fractals.